ESTIMATING THE CONDITION NUMBER OF THE FRECHET DERIVATIVE OF A MATRIX FUNCTION

The Frechet derivative L-f of a matrix function f : C-nxn -> C-nxn is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Frechet derivative and derive upper and lower bounds for it that differ by at most a factor 2. For a wide class of functions we derive an algorithm for estimating the 1-norm condition number that requires O(n(3)) flops given O(n(3)) flops algorithms for evaluating f and L-f; in practice it produces estimates correct to within a factor 6n. Numerical experiments show the new algorithm to be much more reliable than a previous heuristic estimate of conditioning.